# Demand and Elasticity

Business I · Microeconomics · https://tryals.app/learn/business-i/demand-and-elasticity

## Demand Curve & Elasticity

The **demand curve** traces optimal quantity as price varies ($p_x, x^*$), holding income constant. It slopes downward by the **law of demand**, unless the good is a **Giffen good** (an extreme inferior good where income outweighs substitution).

**Price elasticity of demand** measures responsiveness:

$$\varepsilon_p = \frac{\%\Delta Q}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$$

| Value | Label | Revenue when price rises |
|---|---|---|
| $\vert\varepsilon\vert > 1$ | **Elastic** | Falls |
| $\vert\varepsilon\vert = 1$ | **Unit elastic** | Unchanged |
| $\vert\varepsilon\vert < 1$ | **Inelastic** | Rises |

**Determinants**: More substitutes, longer time horizons, luxuries, and larger budget shares all increase elasticity.

> **Pitfall:** Confusing slope ($dQ/dP$) with elasticity. Slope uses units; elasticity uses percentages. Constant slope does NOT mean constant elasticity.

## Elasticity Along a Curve

For $Q = a - bP$, slope is constant at $-b$, but elasticity $\varepsilon = -b \cdot (P/Q)$ changes as $P/Q$ shifts.

- **Top**: High $P$, low $Q$ $\rightarrow$ $|\varepsilon| > 1$ (elastic).
- **Bottom**: Low $P$, high $Q$ $\rightarrow$ $|\varepsilon| < 1$ (inelastic).
- **Midpoint**: $|\varepsilon| = 1$ (unit elastic, revenue peaks).

Total revenue $R = P(a - bP)$ is a parabola, maximised at $P^* = a/(2b)$.

**Cross-price elasticity** measures how good $x$ demand reacts to good $y$ price: $\varepsilon_{xy} = (\%\Delta Q_x)/(\%\Delta P_y)$. Positive for substitutes, negative for complements.

**Income elasticity** is $\varepsilon_M = (\%\Delta Q)/(\%\Delta M)$. Positive for normal goods, negative for inferior goods. **Luxuries** have $\varepsilon_M > 1$.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. If $\varepsilon_p = -0.5$ and price rises 20%, quantity falls by:

**Answer:** 10 (within ±2)

**Why:** $\%\Delta Q = \varepsilon \times \%\Delta P = -0.5 \times 20\% = -10\%$. Inelastic demand: quantity falls by only half the percentage price increase.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/if-p-0-5-and-price-rises-20-quantity-falls-by

### 2. A linear demand curve has a constant slope, yet demand is elastic on its upper half and inelastic on its lower half. How can elasticity change when the slope does not?

A. Because consumers behave differently at high prices for psychological reasons
B. Because elasticity is measured differently above and below the midpoint by convention
C. Because the slope is only approximately constant on a linear curve
D. Because elasticity is a ratio of percentage changes

**Answer:** D. Because elasticity is a ratio of percentage changes

**Why:** Slope is absolute and elasticity is relative. The $P/Q$ factor sweeps from large to small along the curve, which is why total revenue peaks exactly at the midpoint.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/a-linear-demand-curve-has-a-constant-slope-yet-demand-is-elastic-on

### 3. A firm facing elastic demand can increase revenue by raising its price.

**Answer:** False

**Why:** False. With elastic demand quantity falls proportionally *more* than price rises, so revenue drops; it is **inelastic** demand where a price rise raises revenue. This is why cigarette taxes raise revenue and why an airline discounting a leisure route can raise it too, the elasticity decides the direction.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/a-firm-facing-elastic-demand-can-increase-revenue-by-raising-its

### 4. Order the logic of deriving a demand curve.

**Answer:**

1. Fix a price $p_x$
2. Solve utility max: find optimal $x^*(p_x)$
3. Plot the point $(p_x, x^*)$
4. Repeat for different prices
5. Connect points to form the demand curve

**Why:** The demand curve is the trace of optimal choices as price varies, each point is a separate utility-maximisation solution.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/order-the-logic-of-deriving-a-demand-curve

### 5. Match each elasticity type to its formula.

**Answer:**

- Price elasticity → $(\%\Delta Q)/(\%\Delta P)$
- Cross-price elasticity → $(\%\Delta Q_x)/(\%\Delta P_y)$
- Income elasticity → $(\%\Delta Q)/(\%\Delta M)$
- Unit elastic → $|\varepsilon| = 1$: revenue maximised

**Why:** Three elasticity types — price, cross-price, and income — plus the revenue-maximising threshold.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/match-each-elasticity-type-to-its-formula

### 6. Sort by cross-price elasticity sign.

**Answer:**

- Substitutes ($\varepsilon_{xy} > 0$): Butter and margarine, Tea and coffee, Bus tickets and train tickets
- Complements ($\varepsilon_{xy} < 0$): Cars and petrol, Printers and ink cartridges, Tennis rackets and tennis balls

**Why:** Substitutes: price of one rises, demand for the other rises (positive). Complements: price of one rises, demand for the other falls (negative).

Page: https://tryals.app/practice/business-i/demand-and-elasticity/sort-by-cross-price-elasticity-sign

### 7. A rail operator raises fares and its revenue falls. What does that reveal about the elasticity it faces, and where is it on its demand curve?

A. Nothing follows about elasticity; revenue depends on costs as well
B. Elasticity is exactly one, the point where revenue stops rising
C. Demand is elastic: the drop exceeded the price rise
D. Demand is inelastic, since passengers still need to travel

**Answer:** C. Demand is elastic: the drop exceeded the price rise

**Why:** Revenue moving against price is the definition of elastic demand, and on a linear curve that region is the high-price upper half. This is why the revenue test is the practical way to measure elasticity, you observe the direction of the revenue change rather than estimating a curve.

Page: https://tryals.app/practice/business-i/demand-and-elasticity/a-rail-operator-raises-fares-and-its-revenue-falls-what-does-that